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Theorem bdssexg 10962
Description: Bounded version of ssexg 3937. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
bdssexg.bd  |- BOUNDED  A
Assertion
Ref Expression
bdssexg  |-  ( ( A  C_  B  /\  B  e.  C )  ->  A  e.  _V )

Proof of Theorem bdssexg
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 sseq2 3030 . . . 4  |-  ( x  =  B  ->  ( A  C_  x  <->  A  C_  B
) )
21imbi1d 229 . . 3  |-  ( x  =  B  ->  (
( A  C_  x  ->  A  e.  _V )  <->  ( A  C_  B  ->  A  e.  _V ) ) )
3 bdssexg.bd . . . 4  |- BOUNDED  A
4 vex 2613 . . . 4  |-  x  e. 
_V
53, 4bdssex 10960 . . 3  |-  ( A 
C_  x  ->  A  e.  _V )
62, 5vtoclg 2667 . 2  |-  ( B  e.  C  ->  ( A  C_  B  ->  A  e.  _V ) )
76impcom 123 1  |-  ( ( A  C_  B  /\  B  e.  C )  ->  A  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434   _Vcvv 2610    C_ wss 2982  BOUNDED wbdc 10898
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-bdsep 10942
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-in 2988  df-ss 2995  df-bdc 10899
This theorem is referenced by:  bdssexd  10963  bdrabexg  10964  bdunexb  10978
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